Systems will differ between the LCC model - basically point to point and 'legacy' or network airlines.
The DCS (Departure Control Systems) on network airlines are much more complex than LCCs. These will manage (or mis-manage) on-line and interline connections (pax and luggage). Itineraries can be up to (I think) 12 or 14 sectors before manual intervention is required (depends on the airline). It is because of these complexities that a 24 hour period (of checkin for the first flight) is in effect.
International conventions (Warsaw and now Montreal) require all passengers to have both a 'ticket / baggage check' and a boarding card. The LCCs would probably dispense with the boarding card if they could.